Institute for Mathematical Sciences

Preprint ims92-4

E. Cawley

The Teichm\"uller Space of the Standard Action of $SL(2,Z)$ on $T^2$ is Trivial.

Abstract: The group $SL(n,{\bf Z})$ acts linearly on $\R^n$, preserving the integer lattice $\Z^{n} \subset \R^{n}$. The induced
(left) action on the n-torus $\T^{n} = \R^{n}/\Z^{n}$ will be
referred to as the ``standard action''.
It has recently been shown that the standard action of
$SL(n,\Z)$ on $\T^n$, for $n \geq 3$, is both topologically
and smoothly rigid. That is, nearby actions in the space of
representations of $SL(n,\Z)$ into ${\rm Diff}^{+}(\T^{n})$ are
smoothly conjugate to the standard action. In fact, this
rigidity persists for the standard action of a subgroup of
finite index. On the other hand, while the $\Z$ action on
$\T^{n}$ defined by a single hyperbolic element of $SL(n,\Z)$
is topologically rigid, an infinite dimensional space of smooth
conjugacy classes occur in a neighborhood of the linear
action.
The standard action of $SL(2, \Z)$ on $\T^2$ forms an
intermediate case, with different rigidity properties from
either extreme. One can construct continuous deformations of
the standard action to obtain an (arbritrarily near) action to
which it is not topologically conjugate. The purpose of the
present paper is to show that if a nearby action, or more
generally, an action with some mild Anosov properties, is
conjugate to the standard action of $SL(2, \Z)$ on $\T^2$ by a
homeomorphism $h$, then $h$ is smooth. In fact, it will be
shown that this rigidity holds for any non-cyclic subgroup of
$SL(2, \Z)$.